Ordinary Differential Equations

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MR. Sasi Bhusan Padhi

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Core Courses

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Course Name : Ordinary Differential Equations

Code(Credit) : CUTM1514(3-2-1)

Course Objectives

  • To understand most of the physical phenomena from Science and Engineering which are modeled by differential equations
  • To develop the ability to apply differential equations to significant applied and/or theoretical problems.

Learning Outcomes

 

Upon successful completion of this course, students will be able to:

 

  • Learn fundamental concepts of ODE theories and where and how such equations arise in applications to scientific and engineering problems.
  • Be competent in solving linear/non-linear 1st & higher order ODEs using analytical methods to obtain their exact solutions.
  • Find power series solutions of 2nd order differential equations.

Course Syllabus

Module-I    ( T-5 hrs-P-4 hrs-P-2 hrs )

 

Introduction to ordinary differential equations, First order and first degree differential equation, Variable separation method, Homogeneous differential equation, Non-homogeneous equation of first degree.

 

Practice-1  :  Solving One dimensional heat flow equations using Python

Practice-2  :  Solving Predator-Prey Equations using Python

 

Project-1  :  Study on different types of first order and first degree ordinary differential equations used in mathematical models

 

 

Module-II    ( T-6 hrs-P-4 hrs-P-2 hrs )

 

Exact differential equations, Non-exact differential equations, Integrating factors.

 

Practice-3  :  Solving Growth model (exponential case only) using Python

Practice-4  :  Solving Decay model (exponential case only) using Python

 

Project-2  :  Study of Non-exact differential equations and their solutions

 

 

Module-III    ( T-4 hrs-P-4 hrs-P-2 hrs )

 

Linear equation, Bernoulli's equation, Introduction to Second and Higher order Linear Homogeneous and Non-homogeneous equations with constant and variable coefficients.

 

Practice-5  :  Solving Oxygen debt model using Python

Practice-6  :  Solving Mixing Concentrations problems using Python

 

Project -3  :  Applications of first order differential equations to Decay models

 

 

Module-IV    ( T-4 hrs-P-4 hrs-P-0 hrs )

 

Complimentary function and Particular integral, Finding particular integrals for the differential equations containing the Exponential functions, Trigonometric functions (Sine and Cosine) using inverse operator method.

 

Practice-7  :  Solving Escape velocity Problem using Python

Practice-8  :  Solving Free Damped Motion of a Spring System using Python

 

 

 Module-V    ( T-4 hrs-P-4 hrs-P-2 hrs )

 

Finding particular integrals for the differential equations containing Power functions, Product of exponential function with other functions, Functions in the form xV(x)  using inverse operator method, Solving second order linear equations using variation of parameters method.

 

Practice-9  :  Solving Free Undamped Motion of a Spring System using Python

 Practice-10  :  Solving RL circuit problems using Python

 

Project -4  :  Study on  Variation of parameter method for solving second order differential equations

 

 

Module-VI    ( T-4 hrs-P-2 hrs-P-2 hrs )

 

Solving second order linear equations using Method Of Undetermined Coefficients, Linear equations reducible to constant coefficient form, Euler-Cauchy equation, Legendre’s equation.

 

Practice-11  :  Euler Method for solving Second order differential equation using Python

Practice-12  :  Solving Forced Vibration of a Spring System using Python

 

Project -5  :  Study on  Method Of Undetermined Coefficients for solving second order differential equations

 

 

Module-VII     ( T-3 hrs-P-0 hrs-P-2 hrs )

 

Power series solution, Power series solutions of second order differential equations.

 

Project -6  :  Study on Power series solutions

 

 

 

Text Book:

  1. Sinha Roy and S. Padhy, A Course of Ordinary and Partial Differential Equations, Kalyani Publishers, New Delhi.

Chapters: 1, 2(2.1 to 2.7), 4(4.1 to 4.8), 7(7.1 to 7.3)

 

 

Reference Books:

  1. Differential Equations and their Applications: Martin Braun, Springer International.
  2. Advanced Differential Equations: M. D. Raisinghania, S. Chand & Company Ltd., New Delhi.
  3. A First Course in differential Equations with Modeling Applications: G. Dennis Zill, Cengage Learning India Pvt. Ltd.
  4. Text Book of Differential Equations : N.M. Kapoor
  5. Introductory course in Differential Equations : D.A. Murray
  6. Differential Equations: S. L. Ross, John Wiley & Sons, India, 2004.

Session Plan

Session 1

Session 6 & 7

Practice-1

Solving One dimensional heat flow equations using Python

Session 8 & 9

Practice-2

Solving Predator-Prey Equations using Python

Session 10 & 11

Project-1  :  Study on different types of first order and first degree ordinary differential equations used in mathematical models

Session 21 & 22

Practice-3

Growth model (exponential case only) using Python

Session 23 & 24

Practice-4

Decay model (exponential case only) using Python

Session 25 & 26

Project-2  :  Study of  Non-exact differential equations and their solutions

Session 27 & 28

Practice-5

Solving Oxygen debt model problems using Python

Session 29 & 30

Practice-6

Solving Mixing Concentrations problems using Python.

Session 31 & 32

Project-3

Application of first order differential equation to Decay models

Session 33

Second and higher order linear homogeneous differential Equation with constant coefficients

https://www.youtube.com/watch?v=u-ympqH2SBs

https://www.slideshare.net/virajpatel777701/second-order-homogeneous-linear-differential-equations

Session 34

Case-1. Distinct real roots

Case-2. Repeated real roots

Case-3. Conjugate complex roots

https://www.youtube.com/watch?v=jDeLOAbQ8IY

https://www.youtube.com/watch?v=vjp9QC5yy_0

https://www.youtube.com/watch?v=4cFlq1fW_wg

Session 35

Non-homogeneous differential Equation with constant coefficients

https://www.youtube.com/watch?v=wNC3FEnne4o

https://slideplayer.com/slide/6879836/

Session 36

Finding particular integral for an exponential function using inverse operator method

https://www.youtube.com/watch?v=r6mkGRLpm_w

https://www.slideshare.net/sumanmathews/inverse-differential-operator

Session 37

Finding particular integral for Trigonometric functions (Sine and Cosine) using inverse operator method

https://www.youtube.com/watch?v=r6mkGRLpm_w

https://www.slideshare.net/sumanmathews/inverse-differential-operator

Session 38 & 39

Practice-7

Solving Escape velocity Problem using Python

Session 40 & 41

Practice-8

Solving Free Damped Motion of a Spring System using Python

Session 42

Finding particular integral for a power function using inverse operator method

https://www.youtube.com/watch?v=r6mkGRLpm_w

https://www.slideshare.net/sumanmathews/inverse-differential-operator

Session 43

Finding particular integral when exponential function is multiplied with another function using inverse operator method

https://www.youtube.com/watch?v=r6mkGRLpm_w

https://www.slideshare.net/sumanmathews/inverse-differential-operator

Session 44

Finding particular integral when the function is of the form xV(x)  using inverse operator method

https://www.youtube.com/watch?v=r6mkGRLpm_w

Session 45

Solving second order linear equations using variation of parameters method

https://www.youtube.com/watch?v=Ik3YW1JGr_A

Session 46 & 47

Practice-9

Solving Free Undamped Motion of a Spring System using Python

Session 48 & 49

Practice-10

Solving RL circuit problems using Python

Session 50 & 51

Project-4

Study on  Variation of parameter method and Method of   undetermined coeffeciants for solving second order differential equations

Session 52

Solving second order linear equations using method of undetermined coefficients

https://www.youtube.com/watch?v=znE4Nq9NJCQ

https://www.youtube.com/watch?v=2soPqU9H754

Session 53

Equations reducible to linear equations with constant coefficients

https://www.youtube.com/watch?v=ep7xoctAlEY

Session 56 & 57

Practice-11

Euler Method for solving Second order differential equations using Python

Session 58 & 59

Practice-12

Solving Forced Vibration of a Spring System using Python

Session 60 & 61

Project-5

Study on  Method Of Undetermined Coefficients for solving second order differential equations

Session 63

Series solution about an ordinary point

https://www.youtube.com/watch?v=rHzoqkmuWkA

Session 64

Session 65 & 66

Project -6

Study on Power series solutions

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    MR. Sasi Bhusan Padhi

    Asst. Prof. Department of Mathematics
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    MR. Sasi Bhusan Padhi works as an Assistant Professor, Department of Mathematics, Centurion University of Technology and Management. He has received M.Sc degree in Mathematics from Berhampur University in the year 2001 with Gold Medal. He has 17 years of teaching experience in different Graduate and Post Graduate colleges. Now he is doing research in […]